Hyperinvariant subspaces for trace class perturbations of normal operators and decomposability

TL;DR

This paper demonstrates that many trace-class perturbations of diagonalizable normal operators on infinite-dimensional Hilbert spaces possess non-trivial hyperinvariant subspaces, with many being decomposable operators.

Contribution

It establishes the existence of hyperinvariant subspaces for a broad class of trace-class perturbations of normal operators, including many that are decomposable.

Findings

Existence of non-trivial hyperinvariant subspaces for trace-class perturbations.

Many such perturbations are decomposable operators.

Extension of hyperinvariant subspace theory to a large class of perturbed normal operators.

Abstract

We prove that a large class of trace-class perturbations of diagonalizable normal operators on a separable, infinite dimensional complex Hilbert space have non-trivial closed hyperinvariant subspaces. Moreover, a large subclass consists of decomposable operators in the sense of Colojoar\u{a} and Foia\c{s}.